# HG changeset patch # User nipkow # Date 1245754784 -7200 # Node ID b4c8d615bf5d23eb6b59ee46664dce762094364a # Parent 3585bebe49a87887a08b7d28185006423bd5b002# Parent f767c5b1702e4d23d46c52e44ea1313488a51260 merged diff -r f767c5b1702e -r b4c8d615bf5d src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Tue Jun 23 12:58:53 2009 +0200 +++ b/src/HOL/IsaMakefile Tue Jun 23 12:59:44 2009 +0200 @@ -325,7 +325,7 @@ Library/FuncSet.thy Library/Permutations.thy Library/Determinants.thy\ Library/Bit.thy Library/Topology_Euclidean_Space.thy \ Library/Finite_Cartesian_Product.thy \ - Library/FrechetDeriv.thy \ + Library/FrechetDeriv.thy Library/Fraction_Field.thy\ Library/Fundamental_Theorem_Algebra.thy \ Library/Inner_Product.thy Library/Lattice_Syntax.thy \ Library/Legacy_GCD.thy \ diff -r f767c5b1702e -r b4c8d615bf5d src/HOL/Library/Fraction_Field.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Library/Fraction_Field.thy Tue Jun 23 12:59:44 2009 +0200 @@ -0,0 +1,274 @@ +(* Title: Fraction_Field.thy + Author: Amine Chaieb, University of Cambridge +*) + +header{* A formalization of the fraction field of any integral domain + A generalization of Rational.thy from int to any integral domain *} + +theory Fraction_Field +imports Main (* Equiv_Relations Plain *) +begin + +subsection {* General fractions construction *} + +subsubsection {* Construction of the type of fractions *} + +definition fractrel :: "(('a::idom * 'a ) * ('a * 'a)) set" where + "fractrel == {(x, y). snd x \ 0 \ snd y \ 0 \ fst x * snd y = fst y * snd x}" + +lemma fractrel_iff [simp]: + "(x, y) \ fractrel \ snd x \ 0 \ snd y \ 0 \ fst x * snd y = fst y * snd x" + by (simp add: fractrel_def) + +lemma refl_fractrel: "refl_on {x. snd x \ 0} fractrel" + by (auto simp add: refl_on_def fractrel_def) + +lemma sym_fractrel: "sym fractrel" + by (simp add: fractrel_def sym_def) + +lemma trans_fractrel: "trans fractrel" +proof (rule transI, unfold split_paired_all) + fix a b a' b' a'' b'' :: 'a + assume A: "((a, b), (a', b')) \ fractrel" + assume B: "((a', b'), (a'', b'')) \ fractrel" + have "b' * (a * b'') = b'' * (a * b')" by (simp add: mult_ac) + also from A have "a * b' = a' * b" by auto + also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: mult_ac) + also from B have "a' * b'' = a'' * b'" by auto + also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: mult_ac) + finally have "b' * (a * b'') = b' * (a'' * b)" . + moreover from B have "b' \ 0" by auto + ultimately have "a * b'' = a'' * b" by simp + with A B show "((a, b), (a'', b'')) \ fractrel" by auto +qed + +lemma equiv_fractrel: "equiv {x. snd x \ 0} fractrel" + by (rule equiv.intro [OF refl_fractrel sym_fractrel trans_fractrel]) + +lemmas UN_fractrel = UN_equiv_class [OF equiv_fractrel] +lemmas UN_fractrel2 = UN_equiv_class2 [OF equiv_fractrel equiv_fractrel] + +lemma equiv_fractrel_iff [iff]: + assumes "snd x \ 0" and "snd y \ 0" + shows "fractrel `` {x} = fractrel `` {y} \ (x, y) \ fractrel" + by (rule eq_equiv_class_iff, rule equiv_fractrel) (auto simp add: assms) + +typedef 'a fract = "{(x::'a\'a). snd x \ (0::'a::idom)} // fractrel" +proof + have "(0::'a, 1::'a) \ {x. snd x \ 0}" by simp + then show "fractrel `` {(0::'a, 1)} \ {x. snd x \ 0} // fractrel" by (rule quotientI) +qed + +lemma fractrel_in_fract [simp]: "snd x \ 0 \ fractrel `` {x} \ fract" + by (simp add: fract_def quotientI) + +declare Abs_fract_inject [simp] Abs_fract_inverse [simp] + + +subsubsection {* Representation and basic operations *} + +definition + Fract :: "'a::idom \ 'a \ 'a fract" where + [code del]: "Fract a b = Abs_fract (fractrel `` {if b = 0 then (0, 1) else (a, b)})" + +code_datatype Fract + +lemma Fract_cases [case_names Fract, cases type: fract]: + assumes "\a b. q = Fract a b \ b \ 0 \ C" + shows C + using assms by (cases q) (clarsimp simp add: Fract_def fract_def quotient_def) + +lemma Fract_induct [case_names Fract, induct type: fract]: + assumes "\a b. b \ 0 \ P (Fract a b)" + shows "P q" + using assms by (cases q) simp + +lemma eq_fract: + shows "\a b c d. b \ 0 \ d \ 0 \ Fract a b = Fract c d \ a * d = c * b" + and "\a. Fract a 0 = Fract 0 1" + and "\a c. Fract 0 a = Fract 0 c" + by (simp_all add: Fract_def) + +instantiation fract :: (idom) "{comm_ring_1, power}" +begin + +definition + Zero_fract_def [code, code unfold]: "0 = Fract 0 1" + +definition + One_fract_def [code, code unfold]: "1 = Fract 1 1" + +definition + add_fract_def [code del]: + "q + r = Abs_fract (\x \ Rep_fract q. \y \ Rep_fract r. + fractrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})" + +lemma add_fract [simp]: + assumes "b \ (0::'a::idom)" and "d \ 0" + shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)" +proof - + have "(\x y. fractrel``{(fst x * snd y + fst y * snd x, snd x * snd y :: 'a)}) + respects2 fractrel" + apply (rule equiv_fractrel [THEN congruent2_commuteI]) apply (auto simp add: algebra_simps) + unfolding mult_assoc[symmetric] . + with assms show ?thesis by (simp add: Fract_def add_fract_def UN_fractrel2) +qed + +definition + minus_fract_def [code del]: + "- q = Abs_fract (\x \ Rep_fract q. fractrel `` {(- fst x, snd x)})" + +lemma minus_fract [simp, code]: "- Fract a b = Fract (- a) (b::'a::idom)" +proof - + have "(\x. fractrel `` {(- fst x, snd x :: 'a)}) respects fractrel" + by (simp add: congruent_def) + then show ?thesis by (simp add: Fract_def minus_fract_def UN_fractrel) +qed + +lemma minus_fract_cancel [simp]: "Fract (- a) (- b) = Fract a b" + by (cases "b = 0") (simp_all add: eq_fract) + +definition + diff_fract_def [code del]: "q - r = q + - (r::'a fract)" + +lemma diff_fract [simp]: + assumes "b \ 0" and "d \ 0" + shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" + using assms by (simp add: diff_fract_def diff_minus) + +definition + mult_fract_def [code del]: + "q * r = Abs_fract (\x \ Rep_fract q. \y \ Rep_fract r. + fractrel``{(fst x * fst y, snd x * snd y)})" + +lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)" +proof - + have "(\x y. fractrel `` {(fst x * fst y, snd x * snd y :: 'a)}) respects2 fractrel" + apply (rule equiv_fractrel [THEN congruent2_commuteI]) apply (auto simp add: algebra_simps) + unfolding mult_assoc[symmetric] . + then show ?thesis by (simp add: Fract_def mult_fract_def UN_fractrel2) +qed + +lemma mult_fract_cancel: + assumes "c \ 0" + shows "Fract (c * a) (c * b) = Fract a b" +proof - + from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def) + then show ?thesis by (simp add: mult_fract [symmetric]) +qed + +instance proof + fix q r s :: "'a fract" show "(q * r) * s = q * (r * s)" + by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps) +next + fix q r :: "'a fract" show "q * r = r * q" + by (cases q, cases r) (simp add: eq_fract algebra_simps) +next + fix q :: "'a fract" show "1 * q = q" + by (cases q) (simp add: One_fract_def eq_fract) +next + fix q r s :: "'a fract" show "(q + r) + s = q + (r + s)" + by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps) +next + fix q r :: "'a fract" show "q + r = r + q" + by (cases q, cases r) (simp add: eq_fract algebra_simps) +next + fix q :: "'a fract" show "0 + q = q" + by (cases q) (simp add: Zero_fract_def eq_fract) +next + fix q :: "'a fract" show "- q + q = 0" + by (cases q) (simp add: Zero_fract_def eq_fract) +next + fix q r :: "'a fract" show "q - r = q + - r" + by (cases q, cases r) (simp add: eq_fract) +next + fix q r s :: "'a fract" show "(q + r) * s = q * s + r * s" + by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps) +next + show "(0::'a fract) \ 1" by (simp add: Zero_fract_def One_fract_def eq_fract) +qed + +end + +lemma of_nat_fract: "of_nat k = Fract (of_nat k) 1" + by (induct k) (simp_all add: Zero_fract_def One_fract_def) + +lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" + by (rule of_nat_fract [symmetric]) + +lemma fract_collapse [code post]: + "Fract 0 k = 0" + "Fract 1 1 = 1" + "Fract k 0 = 0" + by (cases "k = 0") + (simp_all add: Zero_fract_def One_fract_def eq_fract Fract_def) + +lemma fract_expand [code unfold]: + "0 = Fract 0 1" + "1 = Fract 1 1" + by (simp_all add: fract_collapse) + +lemma Fract_cases_nonzero [case_names Fract 0]: + assumes Fract: "\a b. q = Fract a b \ b \ 0 \ a \ 0 \ C" + assumes 0: "q = 0 \ C" + shows C +proof (cases "q = 0") + case True then show C using 0 by auto +next + case False + then obtain a b where "q = Fract a b" and "b \ 0" by (cases q) auto + moreover with False have "0 \ Fract a b" by simp + with `b \ 0` have "a \ 0" by (simp add: Zero_fract_def eq_fract) + with Fract `q = Fract a b` `b \ 0` show C by auto +qed + + + +subsubsection {* The field of rational numbers *} + +context idom +begin +subclass ring_no_zero_divisors .. +thm mult_eq_0_iff +end + +instantiation fract :: (idom) "{field, division_by_zero}" +begin + +definition + inverse_fract_def [code del]: + "inverse q = Abs_fract (\x \ Rep_fract q. + fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})" + + +lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a" +proof - + have stupid: "\x. (0::'a) = x \ x = 0" by auto + have "(\x. fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x :: 'a)}) respects fractrel" + by (auto simp add: congruent_def stupid algebra_simps) + then show ?thesis by (simp add: Fract_def inverse_fract_def UN_fractrel) +qed + +definition + divide_fract_def [code del]: "q / r = q * inverse (r:: 'a fract)" + +lemma divide_fract [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)" + by (simp add: divide_fract_def) + +instance proof + show "inverse 0 = (0:: 'a fract)" by (simp add: fract_expand) + (simp add: fract_collapse) +next + fix q :: "'a fract" + assume "q \ 0" + then show "inverse q * q = 1" apply (cases q rule: Fract_cases_nonzero) + by (simp_all add: mult_fract inverse_fract fract_expand eq_fract mult_commute) +next + fix q r :: "'a fract" + show "q / r = q * inverse r" by (simp add: divide_fract_def) +qed + +end + + +end \ No newline at end of file diff -r f767c5b1702e -r b4c8d615bf5d src/HOL/Library/Library.thy --- a/src/HOL/Library/Library.thy Tue Jun 23 12:58:53 2009 +0200 +++ b/src/HOL/Library/Library.thy Tue Jun 23 12:59:44 2009 +0200 @@ -25,6 +25,7 @@ Fin_Fun Float Formal_Power_Series + Fraction_Field FrechetDeriv FuncSet Fundamental_Theorem_Algebra